The Øresund Bridge is an official land border and therefore, in addition to vehicle registration papers, a passport should be carried when crossing the bridge (for most EU nationals, a National ID card is sufficient). As both Sweden and Denmark are part of the Schengen Area, border checks are not routine, however they do occur occasionally and you should always carry valid travel documents.
Carrying a proof of purchase (digital or print-out) of the bridge e-vignette can help in case of technical issues at the bridge, but it is not required.
Related articles:
Do I need a payment receipt or print-out of the Øresund Bridge e-vignette?
Do I get a confirmation email for my Øresund Bridge e-vignette order?
Are commercial invoices provided for purchases of the Øresund Bridge online ticket?
How is the Øresund Bridge e-vignette checked?
What vehicles need an Øresund Bridge digital passage ticket?